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Math Expert V
Joined: 02 Sep 2009
Posts: 58473

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Difficulty:   5% (low)

Question Stats: 87% (00:59) correct 13% (01:27) wrong based on 126 sessions

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Circles $$X$$ and $$Y$$ are concentric. If the radius of circle $$X$$ is three times that of circle $$Y$$, what is the probability that a point selected inside circle $$X$$ at random will be outside circle $$Y$$?

A. $$\frac{1}{3}$$
B. $$\frac{\pi}{3}$$
C. $$\frac{\pi}{2}$$
D. $$\frac{5}{6}$$
E. $$\frac{8}{9}$$

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Math Expert V
Joined: 02 Sep 2009
Posts: 58473

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Official Solution:

Circles $$X$$ and $$Y$$ are concentric. If the radius of circle $$X$$ is three times that of circle $$Y$$, what is the probability that a point selected inside circle $$X$$ at random will be outside circle $$Y$$?

A. $$\frac{1}{3}$$
B. $$\frac{\pi}{3}$$
C. $$\frac{\pi}{2}$$
D. $$\frac{5}{6}$$
E. $$\frac{8}{9}$$

We have to find the ratio of the area of the ring around the small circle to the area of the big circle. If $$y$$ is the radius of the smaller circle, then the area of the bigger circle is $$\pi(3y)^2 = 9 \pi y^2$$. The area of the ring $$= \pi(3y)^2 - \pi(y)^2 = 8 \pi y^2$$. The ratio $$= \frac{8}{9}$$.

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Senior Manager  Status: Math is psycho-logical
Joined: 07 Apr 2014
Posts: 402
Location: Netherlands
GMAT Date: 02-11-2015
WE: Psychology and Counseling (Other)

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2
Hey,

Great that we saw how you do it using the actual variables.
However, I just used values.

For the radius of X = 6
For the radius of Y = 2

Then the area for X = 36π
and the area for Y = 4π

32/36 = 8/9.
Intern  B
Joined: 02 Aug 2017
Posts: 6
GMAT 1: 710 Q46 V41 GMAT 2: 600 Q39 V33 Show Tags

I think the easiest way for me was:
$$\frac{πr^2}{π3r^2}$$

Use values Y=1, X=3Y=3

$$1^2 = 1, 3^2=9$$, 1/9 chance it is inside the circle, or 8/9 chance it is outside.
Manager  B
Joined: 10 Sep 2014
Posts: 76
GPA: 3.5
WE: Project Management (Manufacturing)

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1
Bunuel What is concentric? Could you please give a pictorial solution? thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 58473

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Bunuel What is concentric? Could you please give a pictorial solution? thanks

Concentric circles are circles with a common center.
>> !!!

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Intern  B
Joined: 15 Jan 2018
Posts: 12
GMAT 1: 570 Q45 V23 Show Tags

I think this is a high-quality question and the explanation isn't clear enough, please elaborate.
Manager  B
Joined: 20 Jun 2018
Posts: 72
Location: India

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Bunuel wrote:
Official Solution:

Circles $$X$$ and $$Y$$ are concentric. If the radius of circle $$X$$ is three times that of circle $$Y$$, what is the probability that a point selected inside circle $$X$$ at random will be outside circle $$Y$$?

A. $$\frac{1}{3}$$
B. $$\frac{\pi}{3}$$
C. $$\frac{\pi}{2}$$
D. $$\frac{5}{6}$$
E. $$\frac{8}{9}$$

We have to find the ratio of the area of the ring around the small circle to the area of the big circle. If $$y$$ is the radius of the smaller circle, then the area of the bigger circle is $$\pi(3y)^2 = 9 \pi y^2$$. The area of the ring $$= \pi(3y)^2 - \pi(y)^2 = 8 \pi y^2$$. The ratio $$= \frac{8}{9}$$.

How did you deduce from the above question that we need to find the ratio of the areas?
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PLEASE GIVE KUDOS IF YOU WERE ABLE TO SOLVE THE QUESTION. THANK YOU. Re: M22-36   [#permalink] 13 Nov 2018, 21:58
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